This document analyzes the nonlinear dependence and time irreversibility of Mexican oil price behavior using two non-parametric tests: the Hinich portmanteau test and the REVERSE test based on the bispectrum. The results suggest strong evidence of nonlinear structure and time irreversibility in the oil price series, implying that GARCH models cannot fully capture its behavior. Windowed testing also indicates episodic nonlinear dependence, with four windows in the sample period showing nonlinear events. Therefore, the oil price innovations are likely produced by a non-Gaussian distribution rather than being independent and identically distributed.
Garch Models in Value-At-Risk Estimation for REITIJERDJOURNAL
Abstract:- In this study we investigate volatility forecasting of REIT, from January 03, 2007 to November 18, 2016, using four GARCH models (GARCH, EGARCH, GARCH-GJR and APARCH). We examine the performance of these GARCH-type models respectively and backtesting procedures are also conducted to analyze the model adequacy. The empirical results display that when we take estimation of volatility in REIT into account, the EGARCH model, the GARCH-GJR model, and the APARCH model are adequate. Among all these models, GARCH-GJR model especially outperforms others.
This paper empirically examines the role of uncertainty occurred by ‘news’ in Japanese financial markets. A GARCH-MIDAS model is used for estimation. It finds that news-based implied volatility performs well in predicting long-term aggregate market volatilities. A subsample analysis provides that the predictive power of news-based volatility is continuing, as most of the coefficients are positive and significant. So, in general, the news based implied volatility model is associated with high market volatility. Moreover, stock market prices go on rising, different effects that appeared in each subsample period. On the recent period, when Abenomics was conducted, the effect decreased. Also, the effect of exchange rates decrease in short time. When stock prices decrease, volatilities of the stock prices in the past period increase. There is some possibility that markets were too unstable about the movements because of the low prices. Also, the volatility of long-term interest rates increases when the interest rate declines in the recent period under Abenomics. Although interest rates have been quite low in both sample periods, the Bank of Japan (BOJ) started to manage long-term interest rates in the recent period, so market participants seem to begin noticing the movements.
The document discusses three historical figures who led independence movements in Latin America against Spanish rule in the early 19th century: Miguel Hidalgo of Mexico, Simon Bolivar of northern South America, and José de San Martín of southern South America. It prompts the reader to consider which of the three liberators is most influential today and why.
Effect of the process parameters on the surface roughness during magneticIAEME Publication
The document discusses magnetic abrasive finishing (MAF), a process that uses magnetic fields to assist in finishing and deburring of ferromagnetic materials. It summarizes the working principle of MAF and experimental setup used. The study examines the effect of process parameters like abrasive mesh size, applied current, and percentage of iron powder on the surface roughness of stainless steel workpieces. Results from experiments conducted based on a design of experiments approach using different parameter levels are presented in tabular form with percentage improvement in surface roughness as the response.
Garch Models in Value-At-Risk Estimation for REITIJERDJOURNAL
Abstract:- In this study we investigate volatility forecasting of REIT, from January 03, 2007 to November 18, 2016, using four GARCH models (GARCH, EGARCH, GARCH-GJR and APARCH). We examine the performance of these GARCH-type models respectively and backtesting procedures are also conducted to analyze the model adequacy. The empirical results display that when we take estimation of volatility in REIT into account, the EGARCH model, the GARCH-GJR model, and the APARCH model are adequate. Among all these models, GARCH-GJR model especially outperforms others.
This paper empirically examines the role of uncertainty occurred by ‘news’ in Japanese financial markets. A GARCH-MIDAS model is used for estimation. It finds that news-based implied volatility performs well in predicting long-term aggregate market volatilities. A subsample analysis provides that the predictive power of news-based volatility is continuing, as most of the coefficients are positive and significant. So, in general, the news based implied volatility model is associated with high market volatility. Moreover, stock market prices go on rising, different effects that appeared in each subsample period. On the recent period, when Abenomics was conducted, the effect decreased. Also, the effect of exchange rates decrease in short time. When stock prices decrease, volatilities of the stock prices in the past period increase. There is some possibility that markets were too unstable about the movements because of the low prices. Also, the volatility of long-term interest rates increases when the interest rate declines in the recent period under Abenomics. Although interest rates have been quite low in both sample periods, the Bank of Japan (BOJ) started to manage long-term interest rates in the recent period, so market participants seem to begin noticing the movements.
The document discusses three historical figures who led independence movements in Latin America against Spanish rule in the early 19th century: Miguel Hidalgo of Mexico, Simon Bolivar of northern South America, and José de San Martín of southern South America. It prompts the reader to consider which of the three liberators is most influential today and why.
Effect of the process parameters on the surface roughness during magneticIAEME Publication
The document discusses magnetic abrasive finishing (MAF), a process that uses magnetic fields to assist in finishing and deburring of ferromagnetic materials. It summarizes the working principle of MAF and experimental setup used. The study examines the effect of process parameters like abrasive mesh size, applied current, and percentage of iron powder on the surface roughness of stainless steel workpieces. Results from experiments conducted based on a design of experiments approach using different parameter levels are presented in tabular form with percentage improvement in surface roughness as the response.
Relationship between Oil Price, Exchange Rates and Stock Market: An Empirical...IOSRJBM
The theories consider macroeconomic variables to be major determinants of stock market returns or performance. But, the empirical evidence collected from different countries over the world is ambiguous. The effect of macroeconomic variables on stock market has been very popular among the researchers from past many decades. This study chooses two macro variables, i.e., oil price and exchange rate because of their increasing importance nowadays. The variables chosen do not show long run association, while short run association was evident from the analysis.
Spot market of malaysian crude oil adf phillipSyamila Sk
The document analyzes the relationship between spot prices of Malaysian crude palm oil futures and production, stock, and export variables. Unit root tests show the variables are integrated of order one, requiring first differencing to achieve stationarity. Cointegration tests using the Johansen approach show there are significant long-run and short-run relationships between crude palm oil prices and the fundamental variables. The vector error correction model also indicates a strong relationship between prices in the Malaysian futures market and production, stock, and exports that affects prices both ways.
11.association between crude price and stock indicesAlexander Decker
This document summarizes a study that examines the relationship between crude oil prices and three stock market indices (BSE 500, BSE 200, BSE 100) of the Bombay Stock Exchange in India from 2001 to 2011. The study uses various econometric techniques including unit root tests, cointegration tests, vector error correction models, and Granger causality tests. The results show there is a cointegrated long-term relationship between the indices and crude prices. Granger causality tests reveal a one-way causality from the stock indices to crude prices, but not vice versa.
This document summarizes a study that models crude oil prices using a Lévy process. The study finds that a MA(8) model best fits the time series properties of oil price returns. However, there is also evidence of GARCH effects. Therefore, the best overall model is a GARCH(1,1) with errors modeled by a Johnson SU distribution. This hybrid Lévy-GARCH process captures the temporal, spectral and distributional properties of the crude oil price data set.
This paper employs time varying coefficient approach to assess sensitivity of crude oil price change to a number of factors among which change in OPEC crude production and change in US oil production. Our finding indicate crude oil price is inelastic to OPEC production change, with elasticity varying between 0.09 and 0.13, but elastic to US oil production change with elasticity between 0.99 and 1.05. This imply on average crude oil price is about 8 times more responsive to US supply expansion than to OPEC supply decisions. As a result, OPEC producers have a limited impact on oil price reversal but the withdrawal of the US high cost shale technology producers from crude oil production at low price levels can be more effective driver of oil price rises in the future. Such low level sensitivity of oil price to change in OPEC supply imply, other things remain unchanged, for oil price to rise from the current $45 per barrel to $70 per barrel, OPEC cartel needs to cut its current daily production of 27 million barrels by 8 percent.
The Granger causality model is used in the current study to analyze the short-run cause–effect relationship between two stock market indices between 2001 and 2021 using time series data of the daily closing prices of the BSE Sensex and S&P 500 indices listed in the Indian and US stock markets, respectively. The Granger causality model and the augmented Dickey–Fuller test for data stationarity were used in the study to examine the short-term causal link between two market indices during the time period. The outcomes demonstrated the connection between the Indian and US stock markets. The findings imply that both markets have a dynamic, bidirectional relationship. This study provides the investor’s essential inputs for investment decision-making and portfolio diversification. In the current era of globalization, the study is crucial because investors and fund managers now place a high priority on stock market integration. Through fund diversification across equity markets, this study subsequently makes it easier to reduce portfolio risk by providing useful insights on diversification strategies across the stock markets.
Association between crude price and stock indicesAlexander Decker
This study examines the relationship between crude oil prices and stock market indices in India, specifically the BSE 500, BSE 200, and BSE 100, over the period of April 2001 to March 2011. The document reviews previous literature on the relationship between oil prices and stock markets. It then describes the methodology used in the current study, which includes unit root tests, cointegration analysis, and Granger causality tests to analyze the long-term and causal relationships between crude oil prices and the three stock market indices. The results of the empirical analysis show there is a cointegrated long-term relationship between crude prices and each of the three stock market indices, and a one-way causal relationship from the indices to crude prices.
Dynamic asset allocation under regime switching: an in-sample and out-of-samp...Andrea Bartolucci
My work consists of a comparative study of the performances of the multivariate regime switching model against the single regime model in terms of portfolio returns in the context of dynamic asset allocation.
The study was conducted through the practical application, both in-sample and out of-sample, of the two models under various portfolio optimization approaches.
In the first part of the asset allocation exercise I constructed for any asset pricing model, both in-sample and out-of-sample, two dynamic recursive efficient portfolios that maximize the Sharpe among portfolios on the efficient frontier (one with opened budget constraint that permits between 0% and 100% in the riskless asset, one whose weights must sum to 1); in addition short selling, thus negative asset class weights, is not allowed. The other three dynamic recursive portfolios that I constructed have been chosen as those that maximize the investor utility function with three different risk aversion coefficient subject to non-negative weights and opened upper budget constraint.
The second part of the asset allocation exercise focuses only on the out-of-sample period. Here the Copula-Opinion Pooling approach is applied to implement in the asset pricing model views on the asset returns produced by both the single regime model and the regime switching model. The purpose of this section is to investigate and make a comparison of the behavior of the regime switching model and the single state model in the COP framework in terms of both expected and realized portfolio returns and Sharpe ratio in the context of mean-variance and conditional value-atrisk (CVaR) portfolio optimization. Therefore, in addition to the five recursive optimal portfolios chosen with the same portfolio selection process as in the first part, here using conditional value-at-risk as the risk exposure constraint, I derived the dynamic optimal weights of other five different portfolios equally distributed, in terms of CVaR, along the time dependent efficient frontier for different values of the confidence in the views.
The overperformance can be achieved by the more efficient and desirable risk-reward combinations on the state-dependent frontier that can be obtained only by systematically altering portfolio allocations in response to changes in the investment opportunities as the economy switches back and forth among different states. An investor who ignores regimes sits on the unconditional frontier, thus an investor can do better by holding a higher Sharpe ratio portfolio when the low volatility regime prevails. Conversely, when the bad regime occurs, the investor who ignores regimes holds too high a risky asset weight. She would have been better off shifting into the risk-free asset when the bear regime hits. As a consequence, the presence of two regimes and two frontiers means that the regime switching investment opportunity set dominates the investment opportunity set offered by one frontier.
The Use of ARCH and GARCH Models for Estimating and Forecasting Volatility-ru...Ismet Kale
This document discusses volatility modeling using ARCH and GARCH models. It first provides background on ARCH and GARCH models, noting they were developed to model characteristics of financial time series data like volatility clustering and fat tails. It then describes the specific ARCH and GARCH models that will be used in the study, including the ARCH, GARCH, EGARCH, GJR, APARCH, IGARCH, FIGARCH and FIAPARCH models. The document aims to apply these models to daily stock index data from the IMKB 100 to analyze and forecast volatility, and better understand risk in the Turkish market.
Reinvestigating sources of movements in real exchange rateAlexander Decker
This document summarizes a study that investigates the sources of movements in real effective exchange rates (REER) for 5 developing countries from 1975-2010. It applies panel cointegration techniques to test if REER is significantly impacted by changes in real variables like terms of trade, government spending, productivity, trade openness, and capital inflows in the long run. The study finds that REER appreciates in response to improvements in terms of trade, productivity, and capital flows. Trade openness is found to depreciate REER. The results support the view that changes in real variables have a significant influence on REER variations.
This year's SITE Energy Day was devoted to discussing the consequences of oil price fluctuations for markets and actors of the economy. The half-day conference engaged policy-oriented scholars and experts from the business community to discuss the impact of oil price fluctuations on macro fundamentals, international trade, strategies of oil cartels, strategic risk management, and opportunities for change in energy systems.
Matteo Manera, Professor at the University of Milano-Bicocca and Fondazione Eni Enrico Mattei, gave a talk "Asymmetries in the oil-gasoline price relationship".
For more information and research analysis please visit: www.hhs.se/site
This document discusses the evolution of research on the Efficient Market Hypothesis (EMH) in finance. It begins by outlining the three forms of market efficiency put forth in EMH. It then describes how Mandelbrot and others challenged EMH by finding long-term dependence and non-linear relationships in asset price movements, contrary to EMH assumptions of randomness. The document outlines Mandelbrot's rescaled range statistical technique and discusses how later researchers used non-linearity tests to further analyze non-random patterns in markets. It questions the validity of conventional linear tests used to support EMH and argues considering non-linearity is crucial to understanding market efficiency.
This document describes a stochastic volatility model built for the front month Brent oil futures contracts traded on the Intercontinental Exchange in London. It implements a multifactor stochastic volatility model using Bayesian Markov chain Monte Carlo methods. The model is used to forecast conditional volatility and moments, extract risk measures, and could enable option pricing. Summary statistics show the return data exhibits volatility clustering, fat tails, and is stationary.
This paper builds and implements stochastic volatility models for predicting volatility in the Brent oil futures market on the Intercontinental Commodity Exchange in London. Stochastic volatility models describe volatility as having its own stochastic process over time, allowing for applications in derivative pricing, risk assessment, and portfolio management. The paper estimates optimal stochastic volatility models using Bayesian Markov chain Monte Carlo methods and extracts conditional moments, forecasts future volatility, and evaluates model fit. Analysis of stochastic volatility models can provide insight into commodity market behavior and enable more accurate forecasts.
Forecasting Crude Oil Prices by using Deep Learning Based ModelIRJET Journal
This document discusses using deep learning models to forecast crude oil prices. It proposes a new hybrid model that uses deep learning techniques like LSTM, CNN, and RNNs. The model is trained on West Texas Intermediate crude oil market data and shows improved accuracy in price predictions compared to other methods. The document also reviews several other studies applying machine learning and deep learning approaches to crude oil price and energy market forecasting.
This study examines the asymmetric impact of exchange rate changes on stock prices in Germany using monthly data from January 1993 to April 2017. Linear and nonlinear autoregressive distributed lag (ARDL) models are used. The results show that negative changes in the exchange rate (currency devaluation) significantly affect stock prices in the long-run, indicating an asymmetric impact. However, in the short-run, exchange rate changes have a symmetric effect. Diagnostic tests confirm the validity of the nonlinear ARDL model. This empirical evidence suggests that currency devaluations specifically impact German stock prices. The findings provide useful information for policymaking and forecasting the effects of exchange rate fluctuations on the stock market.
The relationship between oil price, exchange rate and islamic stock market in...Alexander Decker
This document summarizes a research journal article that examines the relationship between oil prices, exchange rates, and the Islamic stock market in Malaysia. It provides background on previous research examining the impact of oil price fluctuations on economic factors. The study aims to analyze the dynamic effects of changes in oil prices and exchange rates on Malaysia's FTSE Bursa Malaysia Emas Shariah Index using vector autoregression models. Preliminary findings from monthly data from 2007 to 2011 show the Islamic stock prices are cointegrated with oil prices and exchange rates, with stock prices positively related to oil prices and inversely related to exchange rates.
Oil & Gas Intelligence Report: A Discussion of Price Forecasting MethodolgiesDuff & Phelps
This document from Duff & Phelps analyzes methods for forecasting crude oil prices. It finds that directly predicting prices is difficult due to unpredictable geopolitical and financial factors influencing prices. Instead, it will be more feasible to estimate future price volatility. The document reviews different experts' analyses of historical price trends, finding disagreement on whether prices have historically followed a mean-reverting or random pattern. Duff & Phelps will develop both absolute price and volatility forecasting models, analyzing prices as a geometric Brownian motion with trend or Wiener process.
This document provides a summary of Mauricio Ramirez Grajeda's education and professional experience. It lists that he received a Ph.D from Ohio State University in 2006, has worked as a researcher and professor at the University of Guadalajara since 2008, and is a member of Mexico's National Research System since 2008. It also notes that he has published papers in academic journals and books and taught economics courses at several universities.
Guillermo Sierra Juárez tiene una amplia experiencia académica y profesional en finanzas. Obtuvo doctorados en ciencias financieras y maestrías en economía e investigación de operaciones. Ha trabajado en bancos como Santander y BBVA Bancomer evaluando riesgos. También ha enseñado en varias universidades sobre temas de finanzas cuantitativas.
Relationship between Oil Price, Exchange Rates and Stock Market: An Empirical...IOSRJBM
The theories consider macroeconomic variables to be major determinants of stock market returns or performance. But, the empirical evidence collected from different countries over the world is ambiguous. The effect of macroeconomic variables on stock market has been very popular among the researchers from past many decades. This study chooses two macro variables, i.e., oil price and exchange rate because of their increasing importance nowadays. The variables chosen do not show long run association, while short run association was evident from the analysis.
Spot market of malaysian crude oil adf phillipSyamila Sk
The document analyzes the relationship between spot prices of Malaysian crude palm oil futures and production, stock, and export variables. Unit root tests show the variables are integrated of order one, requiring first differencing to achieve stationarity. Cointegration tests using the Johansen approach show there are significant long-run and short-run relationships between crude palm oil prices and the fundamental variables. The vector error correction model also indicates a strong relationship between prices in the Malaysian futures market and production, stock, and exports that affects prices both ways.
11.association between crude price and stock indicesAlexander Decker
This document summarizes a study that examines the relationship between crude oil prices and three stock market indices (BSE 500, BSE 200, BSE 100) of the Bombay Stock Exchange in India from 2001 to 2011. The study uses various econometric techniques including unit root tests, cointegration tests, vector error correction models, and Granger causality tests. The results show there is a cointegrated long-term relationship between the indices and crude prices. Granger causality tests reveal a one-way causality from the stock indices to crude prices, but not vice versa.
This document summarizes a study that models crude oil prices using a Lévy process. The study finds that a MA(8) model best fits the time series properties of oil price returns. However, there is also evidence of GARCH effects. Therefore, the best overall model is a GARCH(1,1) with errors modeled by a Johnson SU distribution. This hybrid Lévy-GARCH process captures the temporal, spectral and distributional properties of the crude oil price data set.
This paper employs time varying coefficient approach to assess sensitivity of crude oil price change to a number of factors among which change in OPEC crude production and change in US oil production. Our finding indicate crude oil price is inelastic to OPEC production change, with elasticity varying between 0.09 and 0.13, but elastic to US oil production change with elasticity between 0.99 and 1.05. This imply on average crude oil price is about 8 times more responsive to US supply expansion than to OPEC supply decisions. As a result, OPEC producers have a limited impact on oil price reversal but the withdrawal of the US high cost shale technology producers from crude oil production at low price levels can be more effective driver of oil price rises in the future. Such low level sensitivity of oil price to change in OPEC supply imply, other things remain unchanged, for oil price to rise from the current $45 per barrel to $70 per barrel, OPEC cartel needs to cut its current daily production of 27 million barrels by 8 percent.
The Granger causality model is used in the current study to analyze the short-run cause–effect relationship between two stock market indices between 2001 and 2021 using time series data of the daily closing prices of the BSE Sensex and S&P 500 indices listed in the Indian and US stock markets, respectively. The Granger causality model and the augmented Dickey–Fuller test for data stationarity were used in the study to examine the short-term causal link between two market indices during the time period. The outcomes demonstrated the connection between the Indian and US stock markets. The findings imply that both markets have a dynamic, bidirectional relationship. This study provides the investor’s essential inputs for investment decision-making and portfolio diversification. In the current era of globalization, the study is crucial because investors and fund managers now place a high priority on stock market integration. Through fund diversification across equity markets, this study subsequently makes it easier to reduce portfolio risk by providing useful insights on diversification strategies across the stock markets.
Association between crude price and stock indicesAlexander Decker
This study examines the relationship between crude oil prices and stock market indices in India, specifically the BSE 500, BSE 200, and BSE 100, over the period of April 2001 to March 2011. The document reviews previous literature on the relationship between oil prices and stock markets. It then describes the methodology used in the current study, which includes unit root tests, cointegration analysis, and Granger causality tests to analyze the long-term and causal relationships between crude oil prices and the three stock market indices. The results of the empirical analysis show there is a cointegrated long-term relationship between crude prices and each of the three stock market indices, and a one-way causal relationship from the indices to crude prices.
Dynamic asset allocation under regime switching: an in-sample and out-of-samp...Andrea Bartolucci
My work consists of a comparative study of the performances of the multivariate regime switching model against the single regime model in terms of portfolio returns in the context of dynamic asset allocation.
The study was conducted through the practical application, both in-sample and out of-sample, of the two models under various portfolio optimization approaches.
In the first part of the asset allocation exercise I constructed for any asset pricing model, both in-sample and out-of-sample, two dynamic recursive efficient portfolios that maximize the Sharpe among portfolios on the efficient frontier (one with opened budget constraint that permits between 0% and 100% in the riskless asset, one whose weights must sum to 1); in addition short selling, thus negative asset class weights, is not allowed. The other three dynamic recursive portfolios that I constructed have been chosen as those that maximize the investor utility function with three different risk aversion coefficient subject to non-negative weights and opened upper budget constraint.
The second part of the asset allocation exercise focuses only on the out-of-sample period. Here the Copula-Opinion Pooling approach is applied to implement in the asset pricing model views on the asset returns produced by both the single regime model and the regime switching model. The purpose of this section is to investigate and make a comparison of the behavior of the regime switching model and the single state model in the COP framework in terms of both expected and realized portfolio returns and Sharpe ratio in the context of mean-variance and conditional value-atrisk (CVaR) portfolio optimization. Therefore, in addition to the five recursive optimal portfolios chosen with the same portfolio selection process as in the first part, here using conditional value-at-risk as the risk exposure constraint, I derived the dynamic optimal weights of other five different portfolios equally distributed, in terms of CVaR, along the time dependent efficient frontier for different values of the confidence in the views.
The overperformance can be achieved by the more efficient and desirable risk-reward combinations on the state-dependent frontier that can be obtained only by systematically altering portfolio allocations in response to changes in the investment opportunities as the economy switches back and forth among different states. An investor who ignores regimes sits on the unconditional frontier, thus an investor can do better by holding a higher Sharpe ratio portfolio when the low volatility regime prevails. Conversely, when the bad regime occurs, the investor who ignores regimes holds too high a risky asset weight. She would have been better off shifting into the risk-free asset when the bear regime hits. As a consequence, the presence of two regimes and two frontiers means that the regime switching investment opportunity set dominates the investment opportunity set offered by one frontier.
The Use of ARCH and GARCH Models for Estimating and Forecasting Volatility-ru...Ismet Kale
This document discusses volatility modeling using ARCH and GARCH models. It first provides background on ARCH and GARCH models, noting they were developed to model characteristics of financial time series data like volatility clustering and fat tails. It then describes the specific ARCH and GARCH models that will be used in the study, including the ARCH, GARCH, EGARCH, GJR, APARCH, IGARCH, FIGARCH and FIAPARCH models. The document aims to apply these models to daily stock index data from the IMKB 100 to analyze and forecast volatility, and better understand risk in the Turkish market.
Reinvestigating sources of movements in real exchange rateAlexander Decker
This document summarizes a study that investigates the sources of movements in real effective exchange rates (REER) for 5 developing countries from 1975-2010. It applies panel cointegration techniques to test if REER is significantly impacted by changes in real variables like terms of trade, government spending, productivity, trade openness, and capital inflows in the long run. The study finds that REER appreciates in response to improvements in terms of trade, productivity, and capital flows. Trade openness is found to depreciate REER. The results support the view that changes in real variables have a significant influence on REER variations.
This year's SITE Energy Day was devoted to discussing the consequences of oil price fluctuations for markets and actors of the economy. The half-day conference engaged policy-oriented scholars and experts from the business community to discuss the impact of oil price fluctuations on macro fundamentals, international trade, strategies of oil cartels, strategic risk management, and opportunities for change in energy systems.
Matteo Manera, Professor at the University of Milano-Bicocca and Fondazione Eni Enrico Mattei, gave a talk "Asymmetries in the oil-gasoline price relationship".
For more information and research analysis please visit: www.hhs.se/site
This document discusses the evolution of research on the Efficient Market Hypothesis (EMH) in finance. It begins by outlining the three forms of market efficiency put forth in EMH. It then describes how Mandelbrot and others challenged EMH by finding long-term dependence and non-linear relationships in asset price movements, contrary to EMH assumptions of randomness. The document outlines Mandelbrot's rescaled range statistical technique and discusses how later researchers used non-linearity tests to further analyze non-random patterns in markets. It questions the validity of conventional linear tests used to support EMH and argues considering non-linearity is crucial to understanding market efficiency.
This document describes a stochastic volatility model built for the front month Brent oil futures contracts traded on the Intercontinental Exchange in London. It implements a multifactor stochastic volatility model using Bayesian Markov chain Monte Carlo methods. The model is used to forecast conditional volatility and moments, extract risk measures, and could enable option pricing. Summary statistics show the return data exhibits volatility clustering, fat tails, and is stationary.
This paper builds and implements stochastic volatility models for predicting volatility in the Brent oil futures market on the Intercontinental Commodity Exchange in London. Stochastic volatility models describe volatility as having its own stochastic process over time, allowing for applications in derivative pricing, risk assessment, and portfolio management. The paper estimates optimal stochastic volatility models using Bayesian Markov chain Monte Carlo methods and extracts conditional moments, forecasts future volatility, and evaluates model fit. Analysis of stochastic volatility models can provide insight into commodity market behavior and enable more accurate forecasts.
Forecasting Crude Oil Prices by using Deep Learning Based ModelIRJET Journal
This document discusses using deep learning models to forecast crude oil prices. It proposes a new hybrid model that uses deep learning techniques like LSTM, CNN, and RNNs. The model is trained on West Texas Intermediate crude oil market data and shows improved accuracy in price predictions compared to other methods. The document also reviews several other studies applying machine learning and deep learning approaches to crude oil price and energy market forecasting.
This study examines the asymmetric impact of exchange rate changes on stock prices in Germany using monthly data from January 1993 to April 2017. Linear and nonlinear autoregressive distributed lag (ARDL) models are used. The results show that negative changes in the exchange rate (currency devaluation) significantly affect stock prices in the long-run, indicating an asymmetric impact. However, in the short-run, exchange rate changes have a symmetric effect. Diagnostic tests confirm the validity of the nonlinear ARDL model. This empirical evidence suggests that currency devaluations specifically impact German stock prices. The findings provide useful information for policymaking and forecasting the effects of exchange rate fluctuations on the stock market.
The relationship between oil price, exchange rate and islamic stock market in...Alexander Decker
This document summarizes a research journal article that examines the relationship between oil prices, exchange rates, and the Islamic stock market in Malaysia. It provides background on previous research examining the impact of oil price fluctuations on economic factors. The study aims to analyze the dynamic effects of changes in oil prices and exchange rates on Malaysia's FTSE Bursa Malaysia Emas Shariah Index using vector autoregression models. Preliminary findings from monthly data from 2007 to 2011 show the Islamic stock prices are cointegrated with oil prices and exchange rates, with stock prices positively related to oil prices and inversely related to exchange rates.
Oil & Gas Intelligence Report: A Discussion of Price Forecasting MethodolgiesDuff & Phelps
This document from Duff & Phelps analyzes methods for forecasting crude oil prices. It finds that directly predicting prices is difficult due to unpredictable geopolitical and financial factors influencing prices. Instead, it will be more feasible to estimate future price volatility. The document reviews different experts' analyses of historical price trends, finding disagreement on whether prices have historically followed a mean-reverting or random pattern. Duff & Phelps will develop both absolute price and volatility forecasting models, analyzing prices as a geometric Brownian motion with trend or Wiener process.
This document provides a summary of Mauricio Ramirez Grajeda's education and professional experience. It lists that he received a Ph.D from Ohio State University in 2006, has worked as a researcher and professor at the University of Guadalajara since 2008, and is a member of Mexico's National Research System since 2008. It also notes that he has published papers in academic journals and books and taught economics courses at several universities.
Guillermo Sierra Juárez tiene una amplia experiencia académica y profesional en finanzas. Obtuvo doctorados en ciencias financieras y maestrías en economía e investigación de operaciones. Ha trabajado en bancos como Santander y BBVA Bancomer evaluando riesgos. También ha enseñado en varias universidades sobre temas de finanzas cuantitativas.
Willy Cortez es un profesor e investigador en la Universidad de Guadalajara, México. Ha obtenido doctorados en economía de la Universidad de Notre Dame y American University. Ha publicado numerosos artículos y capítulos de libros sobre temas económicos de México. Actualmente dirige la Maestría en Economía en la Universidad de Guadalajara y la Revista EconoQuantum.
Jonas Hedlund is a visiting professor at the University of Guadalajara in Mexico who received his PhD in Economics from the University of Alicante in Spain in 2011. His fields of interest include applied game theory, behavioral economics, and microeconomic theory. He has taught various economics courses in Spain, Sweden, and Mexico and has published papers on topics like communication costs, imitation, and social learning.
El documento presenta el currículum vitae de Rafael Salvador Espinosa Ramírez, profesor e investigador de la Universidad de Guadalajara. Detalla sus estudios de licenciatura, maestría y doctorado, su trayectoria profesional como profesor e investigador, sus distinciones y reconocimientos académicos, sus publicaciones y libros, y su experiencia dirigiendo tesis de posgrado.
La Dra. Valdez es una profesora investigadora en la Universidad de Guadalajara con un doctorado en Economía del Desarrollo de la Universidad de East Anglia en el Reino Unido y una maestría en Economía Regional. Imparte cursos de posgrado y ha trabajado en instituciones de desarrollo internacional como el BCIE. Sus líneas de investigación incluyen el mercado laboral, crecimiento económico y pobreza, con un enfoque en el trabajo infantil y la educación.
Mauricio Ramírez Grajeda is a Mexican economist who received his Ph.D from Ohio State University in 2006. He has since worked as a researcher and professor at Universidad de Guadalajara and has published papers in several academic journals. His research focuses on development economics, spatial econometrics, and analyzing economic phenomena in Mexico like trade, crime, and consumption patterns using quantitative methods.
Este currículum vitae presenta la información personal, educación y experiencia laboral de Leonardo A. Gatica Arreola, profesor asociado en la Universidad de Guadalajara. Detalla su educación formal que incluye un Ph.D. en Economía de la Universidad de Texas en Austin y maestrías del Colegio de México y UNAM. También enumera sus publicaciones, conferencias, trabajos en proceso y experiencia docente en varias universidades mexicanas.
Este currículum vitae resume la experiencia académica y profesional de Guillermo Sierra Juárez. Obtuvo varios grados de posgrado como un doctorado en ciencias financieras y maestrías en economía e investigación de operaciones. Trabajó en bancos como Santander y BBVA Bancomer en cargos relacionados con riesgos financieros. También enseñó en varias universidades sobre temas de finanzas e investigación. Publicó artículos sobre procesos estocásticos y valuación de derivados financieros.
Baruch Ramírez-Rodríguez es un economista empírico con experiencia en diseño y evaluación de programas sociales. Tiene un PhD de la Universidad de East Anglia y experiencia como profesor e investigador en universidades de México y Reino Unido. Se ha desempeñado como consultor para varias agencias gubernamentales mexicanas y ha participado en numerosos proyectos de investigación sobre temas de pobreza, transferencias monetarias y seguridad alimentaria.
El documento presenta el currículum vitae de Antonio Ruiz Porras, profesor-investigador de la Universidad de Guadalajara. Posee estudios de doctorado, maestrías y licenciatura en economía. Ha trabajado como profesor en varias universidades mexicanas y en el Reino Unido. Sus áreas de investigación incluyen finanzas, banca, economía internacional y desarrollo económico. Ha publicado numerosos artículos y capítulos de libros.
Mauricio Ramírez Grajeda is a Mexican economist born in 1970. He received his Ph.D from Ohio State University in 2006. He has since worked as a professor and researcher at various universities in Mexico. His research focuses on development economics, spatial econometrics, and international trade. He has published over 20 papers and book chapters on these topics.
Este documento estudia empíricamente el efecto de las transferencias locales (públicas y privadas) en la probabilidad de alternancia partidista en los municipios de Jalisco, México. Los resultados del modelo de duración muestran que las transferencias privadas evitaron la primera alternancia, mientras que los modelos de panel indican que las transferencias públicas fueron más efectivas para mantener el poder entre administraciones consecutivas. El documento también revisa la literatura sobre el uso estratégico de recursos públicos para mantenerse en el poder.
Este documento presenta un modelo para analizar cómo la distancia ideológica entre partidos políticos afecta negativamente la provisión de bienes públicos. El modelo argumenta que una mayor distancia ideológica entre los partidos y los ciudadanos hace que sea más rentable para los partidos el uso clientelar del empleo gubernamental, asignando más recursos a este fin y menos a bienes públicos. Esto ocurre sin que medie un conflicto social, sino debido a que una mayor distancia ideológica hace menos costoso para los partidos atra
This document analyzes the adequacy of GARCH models for analyzing oil price behavior. It applies two non-parametric tests, the Hinich portmanteau test and REVERSE test, to the Mexican oil price series to explore non-linear dependence and time irreversibility. The results suggest strong evidence of a non-linear structure and time irreversibility, implying that innovations are not independent and identically distributed. The non-linear dependence is also found to be episodic rather than consistent, according to a windowed test. Therefore, the study concludes that GARCH models cannot fully capture the properties of the oil price series.
Este documento presenta un modelo formal para analizar cómo los procesos democráticos electorales y la transparencia afectan la eficiencia gubernamental. A pesar de que la teoría sugiere que estos factores deberían incentivar un mejor desempeño, la evidencia empírica muestra que a menudo los gobiernos siguen siendo ineficientes. El documento explora por qué ocurre esto, concluyendo que incluso con rendición de cuentas y transparencia, existen incentivos para usar el empleo burocrático como patron
Este documento busca identificar episodios de dependencia no lineal en el tipo de cambio mexicano entre 1995 y 2010 utilizando el estadístico de Bicorrelación Hinich Portmanteau. Se encuentran 22 ventanas temporales significativas que rechazan la hipótesis de linealidad, las cuales podrían estar asociadas a eventos políticos y económicos. Los modelos ARCH/GARCH no son adecuados para analizar esta serie debido a que no pueden capturar la dependencia no lineal.
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Este documento presenta un modelo espacial de competencia política para analizar cómo la provisión de bienes públicos y el tamaño de la burocracia se ven afectados por la competencia entre dos partidos políticos. El modelo supone que los partidos compiten en un espacio ideológico unidimensional y que uno de los partidos en el gobierno usa recursos públicos para producir bienes públicos y emplear trabajadores. Los resultados muestran que en cualquier equilibrio político-económico, la burocracia es excesiva y la prov
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Non linear dependence oil price
1. Non-Linear Dependence in Oil Price Behavior
Semei Coronado Ramirez1, Leonardo Gatica Arreola2 and Mauricio Ramirez Grajeda3
1. Department of Quantitative Methods, University of Guadalajara, Zapopan, Jalisco, México
2. Department of Economics, University of Guadalajara, Zapopan, Jalisco, México
3. Department of Quantitative Methods, University of Guadalajara, Zapopan, Jalisco, México
Abstract: In this paper, we analyze the adequacy of GARCH-type models to analyze oil price behavior by applying two
types of non-parametric tests, the Hinich portmanteau test for non-linear dependence and a frequency-dominant test of
time reversibility, the REVERSE test based on the bispectrum, to explore the high-order spectrum properties of the
Mexican oil price series. The results suggest strong evidence of a non-linear structure and time irreversibility. Therefore,
it does not comply with the i.i.d (independent and identically distributed) property. The non-linear dependence, however,
is not consistent throughout the sample period, as indicated by a windowed test, suggesting episodic nonlinear
dependence. The results imply that GARCH models cannot capture the series structure.
Keywords: Bispectrum, time reversibility, nonlinearity, asymmetry, oil price.
1. Introduction consumers. Furthermore, volatility impacts
In recent years, several time series analyses investment behavior in the oil sector. In the
have aimed to understand the behavior of the short run, volatility can also affect storage
crude oil market, particularly its volatility (see demand, the value of firms’ operation options,
for example Refs. [1-5]). and, consequently, the marginal cost of
The application of time-series methods to production [1, 2]. Thus, understanding the price
analyze volatility in economic variables was behavior and volatility of this commodity is an
recently acknowledged by the award of the important issue.
2003 Nobel Prize in economics to Robert Engel Then, a central question is the statistical
and Clive Granger, whose contributions have adequacy of ARCH/GARCH models to analyze
been widely employed in financial time-series oil price behavior. If these formulations are not
models. The simplicity of the linear structures adequate, then any prediction or conclusion
of these types of models lends itself to the study derived from the analysis can be misleading.
of financial asset returns and commodity prices Our goal is to advance in this important
[6-7]. question. Thus, the main aim of this paper is to
The autoregressive conditional explore the oil price behavior and its returns to
heteroskedasticity model (ARCH), and its analyze the adequacy of ARCH/GARCH
generalization GARCH introduced by [8] and specification to study these series, by the
[9] respectively, have been widely applied to application of nonlinearity tests.
model volatility in time series and particularly Since [10] seminal work presented
to model oil price volatility. irrefutable evidence of nonlinear behavior by
This issue is extremely important. the majority of stocks traded on the NYSE,
Volatility is an essential determinant of the studies of this type of behavior on economic
value of commodity-based contingent claims of and financial variables has become a growing
crude oil and of the risk faced by producers and subfield within econometric analysis (see Refs.
[11-16]).
Despite the growing literature that
Corresponding author: Semei Coronado Ramirez, documents the existence of nonlinearity in
PhD., Department of Quantitative Methods, University
of Guadalajara, Periférico Norte 799 esq. Av. José
financial and economic series, most models and
Parres Arias Módulo M 2do. Nivel, Núcleo methods used to analyze financial series,
Universitario Los Belenes, C.P. 45100, Zapopan, particularly their volatility, are based on highly
Jalisco, México. Research fields: time series. E-mail: restrictive statistical assumptions and do not
semeic@gmail.com.
1
2. properly capture the statistical behavior of these (STR-GARCH). This analysis finds that
series. This has been the case for most of the fluctuations in oil prices may be due to the
analyses of the crude oil market (see for nonlinearity of the behavior of different
example Refs. [3, 4, 7-19]). operators in the market [19]. For the Mexican
In this paper, we use the Hinich case, [18] analyze the volatility of Mexican oil
portmanteau bispectrum model to analyze the prices by applying the Generalized
nonlinear and asymmetric behavior of the Autoregressive Conditional Heteroskedasticity
Mexican Maya crude oil price from 1991 to (GARCH) model to study the conditional
2008. We also test for the asymmetric behavior standard deviations and asymmetric effects in
of the series using the REVERSE test. Our the series.
findings suggest that the oil price behavior Comparative analyses of different types of
contains nonlinear structures that cannot be models are also used to examine oil price
captured by any type of ARCH and GARCH behavior. Autoregressive models with
models. We find four windows in the series that Conditional Heteroskedasticity (ARCH),
present nonlinear events. We also reject that the Cointegration, Granger Causality and Vector
series is time reversible. This could be because Autoregressive (VAR) have been compared
the underlying model is nonlinear but the with the Data Mining model to analyze their
innovations are i.i.d. or because the underlying suitability and to obtain information about their
innovations are produced by a non-Gaussian statistical structures. The latter method uses a
probability distribution, although the model is sophisticated statistical tool of mathematical
linear. Therefore, we cannot conclude whether algorithms, fractal mechanics methods, neural
the innovations are i.i.d. networks and decision trees, building on
Analyzing and predicting the price of oil is holistic features to identify variables that
a difficult task due to the random nature of oil determine the fluctuations in oil prices that are
prices. In recent years, studies that attempt to not captured by other models [17].
model oil price behavior have become more Other studies analyze the relationship
sophisticated. In particular, a growing body of between oil prices and other macroeconomic
literature attempts to capture the nonlinear fundamentals, such as GDP, gas and gasoline
behavior of the series. [20] use a methodology prices, interest rate, exchange rate and inflation.
called TEI @ I to analyze the series of monthly [21] use a wavelet spectra method to
crude oil West Texas Intermediate (WTI) prices decompose the oil price series in the time
from 1970 to 2003. This approach decomposes frequency to study how macroeconomic
the series using a different method to model changes affect oil price.
each of the components. It uses an [22] studies the relationship between the
Autoregressive Integrated Moving Average volatility of oil prices and the asymmetry of
(ARIMA) for the linear components that gasoline prices using a VAR model. He
determine the trend, neural networks to concludes that there is a negative relationship
approach the nonlinear behavior incorporated in between oil price volatility and the asymmetry
the error term, and Web-based Tex Mining of gasoline prices.
(WTM) techniques and the Rule-based Expert Other analyses study the relationship
System (RES) to model the non-frequent between oil price and other commodities. [23]
irregular effects. This study examines irregular analyze the behavior of oil prices compared
events in the series and concludes that the series with the prices of sugar and ethanol in Brazil
has a nonlinear behavior with short nonlinear through a TVEECM (Threshold Vector Error
periods affecting the oil price behavior. Correction Models) model. They find evidence
Because it has been observed that oil price of threshold-type nonlinearity, in which the
series present volatility clustering effects, some three commodities have a threshold behavior.
analyses use conditional variance models to Sugar and ethanol are linearly cointegrated, and
parameterize this fact. The relationship between oil prices are determined by the prices of sugar
the nonlinear behavior of the oil price and other and ethanol.
fundamentals has been studied using Smooth Although many of these studies note the
Transition Regression with Generalized existence of nonlinear behavior in the series,
Autoregressive Conditional Heteroskedasticity they do not identify these episodes, and they
2
3. base their analyses on highly restrictive These papers test the adequacy of GARCH
assumptions. However, there is a growing models and detect the nonlinear episodes using
number of analyses of the nonlinear behavior of the Hinich portmanteau model based on the
financial data. With the works of [10] and [24], bicorrelation of the series. [48] developed a
the statistical tools needed to identify the frequency-dominant test of time reversibility
presence of nonlinearity in financial data series based on the bispectrum to explore the high-
have become available [25]. A growing number order spectrum properties. This test provides
of papers analyze episodes of nonlinear information about the time reversibility of the
behavior in financial asset markets. Numerous series; therefore, it is also useful to test the
studies report nonlinearity in the American adequacy of GARCH models. Identifying
market, including [10, 26-32]. Similar findings nonlinear episodes and asymmetric behavior is
have been reported for Asian cases by [14, 33- important for understanding the statistical
37] and for the European markets by [25, 38- characteristics of the oil price time series and its
46]. In the case of Latin American financial volatility, which is the main issue of this paper.
assets, [15] and [47] find nonlinear behavior. To our knowledge, this paper is the first to use
[40] test the validity of specifying a these methods to analyze oil price behavior.
GARCH error structure for financial time-series
data on the pound sterling exchange rate for a 2. Materials and Methods
set of ten currencies. Their results demonstrate
that a structure is statistically present in the data 2. 1 The Hinich Portmanteau Test for
that cannot be captured by a GARCH model or Nonlinearity
any of its variants. [34] study of the Taiwan
Stock Exchange and the stock indices of other
Our nonlinearity analysis is based on the
exchanges, such as New York, London, Tokyo,
Hinich portmanteau model developed by [49].
Hong Kong and Singapore, finds support for
The model separates the series into small, non-
nonlinear behavior in the data series. [36]
overlapping frames or windows of equal length
analyze various international financial indices
and applies the C statistic and the Hinich
to determine the degree of dispersion of the
portmanteau statistic, denoted as H, to test
nonlinearity. They analyze the Taiwan stock
whether the observations in each window are
market to determine whether the phenomenon is
white noise.
truly characteristic of financial time series.
Their results indicate that nonlinearity is, in Let x(t) denote the time series where t is
fact, universal among such series and is found an integer, t = 1,2,3,..., which denotes the time
in all studied markets and the vast majority of unit. The series is separated into non-
stocks traded on the Taiwanese exchange. [32] overlapping windows of length n. The kth
analyzes 60 stocks on the NYSE that represent {
window is x(tk ),x(tk +1),...x(tk + n-1) and }
companies with varying market capitalizations the next non-overlapping window is
{ x(t ),x(tk+1 +1),...x(tk+1 + n-1)} ,
for odd years between 1993 and 2001. The
results show a significant statistical difference k+1
in the level and incidence of nonlinear behavior where tk+1 = tk + n . For each window, the null
( )
among portfolios of different capitalization
categories. Highly capitalized stocks show the hypothesis is that x tk is a stationary pure
greatest levels and frequency of nonlinearity, noise process with zero bicorrelation, and the
followed by medium and thinly capitalized
stocks. These differences were more
( )
alternative hypothesis is that x tk is a random
pronounced at the beginning of the 1990s, but process for each window with correlation not
they remain significant for the entire period. equal to zero, Cxx (r ) = E é x(t)x(t + r ) ù , or non
ë û
Nonlinear correlation increased over the course zero bicorrelation,
of the decade under study for all portfolios, Cxxx (r ,s) = E é x(t)x(t + r )x(t + s) ù , in the
whereas linear correlation declined. There were ë û
also cases of sporadic correlation among the primary domain 0 < r < s< L , where L is the
portfolios, suggesting that the relationship is number of lags defined in each window.
more dynamic than was previously thought.
3
4. We now consider the
standardized asymptotic theory (see Ref. [50]). If the C and
( )
observations, z tk , with z tk = ( ) ( )
x tk - m x
,
H statistics reject the null for pure noise for the
data generated by (6), then the structure of the
2
sx series cannot be modeled by an ARCH,
where m x is the expected value of the process GARCH or other stochastic volatility model.
and s x is the variance. Then, the sample
2
2.2 Testing for Reversibility
correlation is given by the following:
1 n-r
å Z(t)Z(t + r ) .
Our second approach is the analysis of the
Czz (r ) = (1)
n- r t=1 statistical structure of the series cycle by testing
Therefore, the C test statistic is as follows: for time reversibility. If the time series is i.i.d.
L forward and backward, then time is said to be
C = å (Czz (r ))2 ~ c L .
2
(2) reversible; otherwise, it is irreversible.
r =1 As in the case of the business cycle, we
( )
The r ,s sample bicorrelation is given by the expect that the oil price cycles will be
asymmetric due to their fundamentals.
following:
Therefore, the impulse response functions
1 n-s
Cxxx (r ,s) = å Z(t )Z(t + r )Z(t + s) , (3)
n- s t=1
cannot be invariant, and the commonly used
models cannot capture this. [50] developed a
for 0 £ r £ s. frequency-domain test of time reversibility
The H statistic tests for the existence of based on the bispectrum called the REVERSE
non-zero bicorrelation in the sample windows test. Similar to the TR test of [51], the
and is distributed in the following way: REVERSE test examines the behavior of
L s-1 estimated third-order moments; however, it has
H = å å Gzzz (r ,s) ~ c (2L-1)( L/2)
2
(4) a better analysis of variance and higher power
s=2 r =1 to test against time-irreversible alternatives.
with G(r ,s) = n- sCzzz (r ,s) . The number of
If x(t) represents a third-order stationary
lags is defined by L n , with 0 c 0.5 .
c
process with mean zero, then the third-order
Based on the results of [49], the recommended moment is defined by the following:
value for c is 0.4. A window is significant for cx (r, s) = E[ x(t)x(t + r )x(t + s)],
any of the statistical C or H if the null (6)
hypothesis is rejected at a significant threshold s £ r, r = 0,1,2,...
level. For each of the two tests for The bispectrum is a double Fourier
autocorrelation and bicorrelation, the for transformation of the third-order cumulative
each window is a = 1- é(1- a c )(1- a H ) ù (see
ë û function. If the bispectrum is defined by
Ref. [34]). In this study, we use a threshold of frequencies f 1 and f 2 in the domain,
0.1 percent. W = {( f1, f2 ) : 0 < f1 < 0.5, f2 < f1,2 f1 + f2 <1} , (7)
Examining whether ARCH, GARCH or
then the bispectrum is defined as follows:
any other volatility stochastic model can ¥ ¥
adequately characterize the series using the
above test can be done by transforming the
Bx ( f1, f2 ) = å å c (r,s)exp[ -i2p ( f r + f s)] .
x 1 2 (8)
t1 =-¥ t2 =-¥
returns into a set of binary data: If x(t) is time reversible, then
ì1 if z(t) ³ 0 cx (r,s) = cx (-r,-s) ; thus, the imaginary part of
[ x(t)] = í-1 if otherwise . (5)
î the bispectrum is zero. More elaboration on the
imaginary part can be found in the work of [53].
If z t is generated by an ARCH, We divide the sample
GARCH or stochastic volatility process with {x(0), x(1),..., x(T -1)} within each non-
innovation symmetrically distributed around a overlapping window of length Q and define the
zero mean, then the binary transformed data (5) discrete Fourier transformation as fk = k / Q . If
converts into a Bernoulli process [14] with
well-behaved moments with respect to the T is not divisible by Q, then T is the sample size
of the last window, with some data not used.
4
5. The number of frames used is equal to
P = [T / Q] , where the brackets signify the
(
If the imaginary part Im Bx f1 , f2 = 0 , )
then the REVERSE statistic is distributed c 2
division of an integer. The resolution bandwidth
() is defined as d = 1/ Q. with M = T 2 /16 degrees of freedom [51].
( )
This test can be also used for nonlinear
Let Bx f k , f k be the smoothing time series to detect deviations in the series
1 2
Bx ( f1 , f2 ) ,
under the assumption of Gaussianity [53].
estimator for which obtains If the null hypothesis of time reversibility
( ) from the average of over values
Bx f k , f k
1 2
is rejected, then the series may be time
irreversible in two ways. The underlying model
Y( f , f )
could be nonlinear while the innovations are
for
k1 k2
across the P frames, where symmetrically distributed. The second
Q alternative is that the underlying innovations
Y( fk1 , fk2 ) = X( fk1 )X( fk2 )X *( fk2 + fk2 ), (9) come from a non-Gaussian probability
distribution, and the model is linear. Hence, the
and
Q-1
REVERSE is not equivalent to a nonlinearity
X( fk ) = å x(t + (p.Q)exp [-i 2p fk (t + (p.Q))] (10) test [54].
t=0
for the pth frames of length Q, for 3. Results and Discussion
p = 0,1,..., P-1.
[48] show that if the sequence (f ,f )
k1 k2
The data used in this analysis were
obtained from the Economatica database. The
converges to ( f , f ), this is a consistent and
1 2
series is the daily Mexican Maya crude oil price
asymptotically normal estimator of the from 01/01/1991 to 08/28/2008, denominated in
bispectrum Bx ( f1, f2 ) . Then, the large sample U.S. dollars. The series has a total of 4,607
observations. Figure 1 shows the behavior of
variance of Bx f k , f k ( 1 2
) is as follows: the Maya oil spot price during the analyzed
period.
æ ö
Var = ç 2 ÷ × Sx f k
1
( ) S (f ) S (f + fk ) , (11)
è ( )
çdT ÷
ø
1 x k2 x k1 2 Figure 1. Maya oil prices for the period 1/01/91-
08/28/08 in U.S. dollars.
where Sx ( f ) is defined as a consistent 140
estimator with an asymptotic normal 120
distribution of the frequency spectrum f, and δ 100
is the resolution bandwidth set in the
calculation. 80
The normalized estimator of the 60
bispectrum is the following:
40
A( fk1 , fk2 ) = P /T × Bx ( fk1 , fk2 ) /Var 1/2 . (12)
20
The imaginary part of A( fk1 , fk2 ) is
0
denoted by Im A( fk1 , fk2 ) . Then, the statistical 1000 2000 3000 4000
REVERSE is represented below:
Before applying the different tests in our
å å Im A( f , f
2
REVERSE = k1 k2 ) (13)
analysis, the data were transformed to the
compounded returns series by the following
(k1 ,k2 )ÎD
relationship:
where
æ p ö
D= {( k ,k ) : ( f , f ) ÎW} .
1 2 k1 k2
(14) P = ln ç t ÷ ,
t
è pt-1 ø
5
6. where pt is the closing price at time t. Figure 2
shows the behavior of the logarithmic returns of Table 2 presents the C and H statistics
the Mayan oil price for the analyzed period. results for the binary transformation of the full
range. A 0.1% threshold was used for the p-
values of the Hinich portmanteau test. The null
hypothesis of pure noise is clearly rejected. In
both cases, for statistics C and H, the p-value is
practically zero. Thus, it may be inferred that
Figure 2. Logarithmic returns of Maya oil prices for they are characterized by nonlinear
the period 01/01/91-08/28/08. dependencies, which contradicts the assumption
2.0
of independent and identical distributed
innovations.
1.8
Thus, GARCH models are not suitable to
1.6
capture the statistical structure of the underlying
1.4 process.
1.2
1.0 Table 2. C, H and REVERSE statistics for the entire
period transformed
0.8
Period 01/01/91-08/28/08
0.6
Number of observations 4607
0.4 Number of lags 29
1000 2000 3000 4000
p-value of C 0.000
p-value of H 0.000
3.1 Results
The summary of statistics for the Mexican To further explore whether nonlinear
Mayan oil price returns series is documented in dependence is present throughout the full
Table 1. It is apparent that the return over the sample or within certain sub-periods, we divide
complete series is positive and quite large the series into a set of 117 non-overlapping
because the mean is 1. The median is also 1, but windows with 30 observations each and analyze
skewness is positive. Kurtosis is also positive them. This process helps to clarify the nature of
and extremely large; therefore, the distribution market efficiency over different periods. The
has a leptokurtic shape. This does not mean that length of the windows should be long enough to
the shape of the distribution has less variance, apply statistical C and H but short enough to
but it is more likely that this distribution offers capture nonlinear events within each window
larger extreme values than a normal [40]. We use a length of 30 observations
distribution. The positive skewness and the high because a month lasts 30 days, on average.
kurtosis values imply deviations from For both the C and H statistics, we use a
Gaussianity in the series [56]. threshold of 0.1 percent. The results of the C
Finally, as expected, the Jarque-Bera and H tests are shown in Table 3.
normality test statistic is quite large, and the
Table 3. Windows test results
null hypothesis of normality is rejected.
Threshold 0.001
Table 1. Summary statistics for Maya oil price # of Windows 135
returns over the period 01/01/91-08/28/08 Length of Window 30
Number of Observations 4,607 # Windows sig. C 1
# Windows sig. H 19
Mean 1
% Windows C 0.740
Median 1
% Windows H 14.070
Standard Deviation 0.03
Skewness 7.21 p-value of REVERSE 0.000
Kurtosis 184.62
Jarque-Bera Test Statistic 6371923 Given the chosen threshold of 0.01, the
p-Value 0.00 results show that the C statistic rejects the null
hypothesis of pure noise in a single window.
6
7. However, with the H statistic, we found 19 periods of pure noise. To complement this
significant windows. These results show that evidence, the REVERSE test showed that the
the percentage of significant C and H windows series was not time reversible and did not
is low. These significant windows reject the comply with the property that the innovations
null hypothesis of pure noise, indicating the are i.i.d.
presence of nonlinearity confined to these
windows. Although the tests find a single C Our results indicates that GARCH models
window, it is sufficient to influence the overall fail to capture the data generating process for
performance of the oil price. This peculiarity the Mexican oil returns.
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